Page 121 - Probability, Random Variables and Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES

z > 0

Since fxyz(x, y, z) = fx(x) fy(y) fz(z), X, Y, and Z are independent.

3.45. Show that
fxuz(x9 Y, 4 'f&, Y(Z x, Y)~Y,X(Y x)fx(x)
I
I
By definition (3.79),

Hence
Now, by Eq. (3.38),
fxu(x9 Y) = fYIX(Y I x)fx(x)
Substituting this expression into Eq. (3.100), we obtain

SPECIAL DISTRIBUTIONS
3.46. Derive Eq. (3.87).
Consider a sequence of n independent multinomial trials. Let Ai (i = 1, 2, . . . , k) be the outcome of a
single trial. The r.v. Xi is equal to the number of times Ai occurs in the n trials. If x,, x,, .. ., x, are
nonnegative integers such that their sum equals n, then for such a sequence the probability that Ai occurs xi
times, i = 1, 2, .. ., k-that is, P(X1 = x,, X, = x,, . . ., X, = x,tcan be obtained by counting the number
of sequences containing exactly x, A,'s, x, A,'s, . . . , x, A,'s and multiplying by p,x1p2x2 . . . pkxk. The total
number of such sequences is given by the number of ways we could lay out in a row n things, of which x,
are of one kind, x, are of a second kind, . . . , x, are of a kth kind. The number of ways we could choose x,
)
positions for the Alls is ; after having put the Al's in their position, the number of ways we could
choose positions for the A,'s is (n i2x1), and so on, Thus, the total number of sequences with x, A19s, x,
A,'s, . . . , xk Ah's is given by

((n ,xl)( - ;3- x2) .*. (n - x1 - x2 - - Xk-1 )
Xk
n ! . . . (n-x, -x,--..- xk- I)!
-
- (n - x,)!
x,!(n - x,)! x,!(n - x, - x,)! x,! O!
- n !
-
x1!x2! --- x,!
Thus, we obtain

3.47. Suppose that a fair die is rolled seven times. Find the probability that 1 and 2 dots appear twice
each; 3,4, and 5 dots once each; and 6 dots not at all.
Let (XI, X,, . . . , X,) be a six-dimensional random vector, where Xi denotes the number of times i dots
appear in seven rolls of a fair die. Then (X,, X, , . . . , X,) is a multinomial r.v. with parameters (7, p,, p,, . . . ,